I am new here. I was reading another post here and this got me wondering what can be said about the size of the following kl divergence neighborhoods. Consider these two kl-divergence neighbourhood for a fixed distribution $P'$ and some $\alpha \geq 0$ $$ \mathbf{P} = \{P : D_{KL}(P||P') < \alpha\}\\ \mathbf{Q} = \{Q : D_{KL}(P'||Q) < \alpha\} $$

I am wondering anything can be said about $|\mathbf{P }|$ and $ |\mathbf{ Q}|$. Because the KL-divergence is asymmetric, I doubt they are not equal and the answer seems to be dependent on $P'$. When $\alpha=0$, the problem is trivial. I am curious about when $\alpha > 0$. I hope this question made sense. Thanks.